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An idealized self-gravitating interior with constant mass density. Negligible surface pressure and spherical hydrostatic equilibrium give $M=4\pi \rho R^3/3$ and the uniform-density planetary pressure profile. It is the $n=0$ polytrope interpretation, rather than a literal substitution into $P=K{\rho }^{1+1/n}$. It approximates weak compression and misses real compositional stratification.

For an incompressible planetary interior of density $\rho$, the enclosed mass and local gravity are $M(r)=4\pi \rho r^3/3$ and $g(r)=4\pi G\rho r/3$. Hydrostatic equilibrium therefore givesIntegrating inward from negligible surface pressure at $r=R_p$ yields the uniform-density planetary pressure profile:The surface gravity is $g_s=GM/R_p^2=4\pi G\rho R_p/3$. Eliminate $\rho R_p$ to obtainThe units of $g_s^2/G$ are pressure. For a nonzero imposed surface pressure, add $P_s$ throughout and interpret the boxed central value as $P_c-P_s$. The constant-density approximation is crucial; real centrally concentrated rocky planets need a different profile and generally a larger central pressure at the same mass and radius.